gearbox fault diagnosis using adaptive wavelet filter

Mechanical Systems and Signal Processing (2003) 17(6), 1259–1269

doi:10.1006/mssp.2002.1507

GEARBOX FAULTDIAGNOSISUSINGADAPTIVEWAVELETFILTER

J. Lin and M. J. Zuo

Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8E-mail: [email protected]

(Received 17 December 2001, accepted after revisions 21 June 2002)

Vibration signals from a gearbox are usually noisy. As a result, it is difficult to find earlysymptoms of a potential failure in a gearbox. Wavelet transform is a powerful tool todisclose transient information in signals. An adaptive wavelet filter based on Morletwavelet is introduced in this paper. The parameters in the Morlet wavelet function areoptimised based on the kurtosis maximisation principle. The wavelet used is adaptivebecause the parameters are not fixed. The adaptive wavelet filter is found to be veryeffective in detection of symptoms from vibration signals of a gearbox with early fatiguetooth crack. Two types of discrete wavelet transform (DWT), the decimated with DB4wavelet and the undecimated with harmonic wavelet, are also used to analyse the samesignals for comparison. No periodic impulses appear on any scale in either DWTdecomposition.

# 2003 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Gearboxes are widely used in industrial applications. An unexpected failure of the gearboxmay cause significant economic losses. It is, therefore, very important to find early faultsymptoms from gearboxes. Tooth breakage is the most serious failure for a gearbox. Earlydetection of cracks in gears is essential for prevention of sudden tooth breakage. Usually,vibration signals are acquired from accelerometers mounted on the outer surface of abearing case. The signals include vibrations from the meshing gears, shafts, bearings, andother parts. Thus, the signals are always complex and it is difficult to diagnose a gearboxfrom such vibration signals.

Misalignment of the shafts can be detected using sidebands [1]. Sidebands can be easilydetected in the frequency domain since they are composed of only a few single-frequencycomponents. As for crack recognition, the key is to find periodic impulses in signals [1].Impulses are short in time duration and usually hidden in noises unless the crack is verybig. Phase demodulation based on synchronous time average has been considered to be themost effective technique for crack detection in a gear [2]. Synchronous time average is aneffective way to remove noises from periodic signals. Because the signal of interest isperiodic and noises are random, the noises will tend to zero when the number of periodstends to infinity. However, the synchronous time-average technique needs a referencesignal to ensure synchronisation and thus, a proximity probe should be used to obtain anaccurate reference signal. Since it is often impossible to obtain such reference signals inpractical applications, new techniques are needed to analyse signals for crack detection ingearboxes.

Wavelet analysis has been successfully used in non-stationary vibration signalprocessing and fault diagnosis [2–6]. Wavelet analysis is capable of providing both time-

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J. LIN AND M. J. ZUO1260

domain information and frequency-domain information simultaneously. Similar to awavelet function, the transient feature components of vibration signals have local energydistributions in both the time domain and the frequency domain. Wavelet functions can beused for detection of transient feature components because they have similar time–frequency structures. Since different types of wavelets have different time–frequencystructures, we should use the wavelet whose time–frequency structure matches that of thetransient component the best in order to detect the transient component effectively.Discrete wavelet transform (DWT) uses dyadic discretisation. The structures of both thedecimated DWT and the undecimated DWT are too rigid for them to provide a goodmatch to the structure of the transient component. During the last few years, continuouswavelet transform (CWT) has been used for gearbox diagnosis [2, 5]. However, the methodused in [2] still requires a reference signal and the one mentioned in [5] has to select thethreshold manually based on experience. An adaptive method should be set up toovercome those deficiencies.

In this paper, an adaptive wavelet filter based on Morlet wavelet is proposed.The shape of the wavelet filter (time–frequency resolution) is automatically adjustedbased on the kurtosis maximisation rule. This filter is used to extract periodicimpulses immersed in noisy signals for gear crack recognition. When the time series ofthe wavelet are convoluted with the signal, the filtered result is obtained. BecauseMorlet wavelet is a cosine function with exponential decay on both sides and is very muchlike an impulse, it is suitable for impulse detection. To get good performance of thefiltering result, the scale and the time–frequency balance parameter for Morlet waveletmust be selected carefully. An adaptive method based on kurtosis maximisation isproposed for this task. We find that it is very effective for detecting periodic impulseshidden in noises.

This paper is organised as follows. The background introduction is given in Section 1. InSection 2, the adaptive Morlet wavelet filter is set up. Kurtosis maximisation is used forselection of the wavelet filter parameters. In Section 3, two types of DWT, the decimatedone and the undecimated one, are used to detect fault symptoms for vibration signals froma gearbox with fatigue cracks on a gear. No abnormal features are found by eithermethod. In Section 4, the adaptive wavelet filter is used for gear crack recognition. Theperiodic impulses are detected successfully by using this method. Conclusions are given inSection 5.

2. ADAPTIVE MORLET WAVELET FILTER

2.1. WAVELET TRANSFORM

Wavelet transforms are inner products between signals and the wavelet family, whichare derived from the mother wavelet by dilation and translation. Let cðtÞ be the motherwavelet, the daughter wavelet will be ca;bðtÞ ¼ cððt� bÞ=aÞ, where a is the scale parameterand b is the time translation. By varying the parameters a and b, we can obtain differentdaughter wavelets that constitute a wavelet family. Wavelet transform is to perform thefollowing operation:

Wða; bÞ ¼1ffiffiffia

pZ

xðtÞcn

a;bðtÞ dt ð1Þ

where ‘*’ stands for complex conjugation.Many research results have been published on wavelet reconstruction. Early research

focused on orthogonal wavelet reconstruction. It is rather easy to perform inverse wavelettransform for orthogonal wavelet. The latest research focuses more on non-orthogonal

GEARBOX FAULT DIAGNOSIS 1261

wavelet reconstruction [7]. According to the original definition of wavelet transform, thereis a universal reconstruction equation for any type of wavelet:

xðtÞ ¼ C�1c

Z ZWða; bÞca;bðtÞ

da

a2db ð2Þ

where

Cc ¼Z 1

�1j #ccðoÞj2=joj do51 ð3Þ

#ccðoÞ ¼Z

cðtÞ expð�jotÞ dt: ð4Þ

If a daughter wavelet is viewed as a filter, wavelet transform is simply a filteringoperation. Usually, we attempt to discover feature signals by reconstructing the waveletcoefficients at selected scales or the wavelet coefficients shrunk through certain methods.To do this, we have to have prior information on the signal to be identified.

2.2. ADAPTIVE MORLET WAVELET FILTER

Morlet wavelet is one of the most popular non-orthogonal wavelets. The definition ofMorlet is

cðtÞ ¼ expð�b2t2=2Þ cosðptÞ: ð5Þ

It is a cosine signal that decays exponentially on both the left and the right sides. Thisfeature makes it very similar to an impulse. It has been used for impulse isolation andmechanical fault diagnosis through the performance of a wavelet de-noising procedure [5].However, in practice, it is not easy to provide a proper threshold for wavelet de-noising.We can avoid this by using an adaptive wavelet filter instead of wavelet de-noising.

The impulses that exist in vibration signals are usually different from theoreticalimpulses. Theoretical impulses have even energy distribution in the frequency domain. Asimulated impulse that often exists in vibration signals is shown in Fig. 1(a). Thecorresponding frequency spectrum is shown in Fig. 1(b). Of course, such impulses areusually immersed in noisy signals. It can be easily seen from Fig. 1 that the frequencyspectrum of the impulse does not have an even distribution. The main energy focuses on aspecific frequency band and decays quickly with the increase and the decrease of thefrequency, which is unlike that of a theoretical impulse. In addition, the decaying rates aredifferent for the left and the right sides.

A daughter Morlet wavelet is obtained by time translation and scale dilation from themother wavelet, as shown in the following formula [7]:

ca;bðtÞ ¼ ct� b

a

� �¼ exp �

b2ðt� bÞ2

2a2

� �cos

pðt� bÞa

� �ð7Þ

where a is the scale parameter for dilation and b is the time translation. It can also belooked at as a filter.

To identify the immersed impulses by filtering, the location and the shape of thefrequency band corresponding to the impulses must be determined first. Scale a andparameter b control the location and the shape of the daughter Morlet wavelet,respectively. As a result, an adaptive wavelet filter could be built by optimising the twoparameters for a daughter wavelet.

Several researchers have reported on how to select the mother wavelet that adapts thebest to the signal to be isolated [2, 7–11]. Details on how to select b in Morlet wavelet tomake the mother wavelet match the signal to be isolated are provided in [5]. In this paper,

0 100 200 300 400 500 600 700 800 900 1000-0.8

-0.6

-0.4

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0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400 450 500-14

-12

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dB

Frequency

Time

Am

plitu

de

(a)

(b)

Figure 1. A simulated impulse. (a) The waveform of a simulated impulse. (b) The frequency spectrum of thesimulated impulse.


we focus on finding the best wavelet filter (the daughter wavelet of a Morlet wavelet)instead of optimal wavelet reconstruction.

Wang has studied the differences between single- and double-sided Morlet wavelets [6].Their frequency spectra are quite different. Since a real impulse is usually non-symmetric,we choose to use the right-hand side of Morlet wavelet as the basis. Such wavelets shouldmatch the behaviours of hidden impulses the best.

Kurtosis is used in engineering for detection of fault symptoms because it is sensitive tosharp variant structures, such as impulses [12]. The bigger the impulse in signals, the largerthe kurtosis. As a result, kurtosis can be used as the performance measure of a Morlet


wavelet filter. The definition of kurtosis is

kurtðyÞ ¼ Eðy4Þ � 3½Eðy2Þ�2 ð6Þ

where y is the sampled time series and E represents the mathematical expectation of theseries.

The procedure to perform the adaptive wavelet filtering is as follows:

(1) Vary the parameters a and b within preselected intervals to produce different daughterwavelets.

(2) Perform wavelet filtering using each daughter wavelet and calculate the kurtosis ofeach outcome.

(3) Compare the kurtosis value. The parameters a and b that correspond to the largestkurtosis are the best parameters to use to reveal the hidden impulses.

3. GEARBOX VIBRATION SIGNAL PROCESSING USING DWT

Gearbox diagnosis is challenging in engineering because the vibration signals from agearbox are complex. They usually represent vibrations from many different sources.Feature components are hidden among many irrelevant components. To diagnose thegearbox faults effectively, the most important thing is to isolate the feature componentsfrom original complex signals.

Two types of feature components are often encountered when faults occur. The first oneis sideband, which usually indicates the faults relevant to misalignment. The second one isperiodic impulse, which usually indicates tooth crack or even tooth breakage. Tooth crackmust be detected early to prevent tooth breakage.

The gearbox in our experiment is illustrated in Fig. 2. Its transmission path is as follows.

input !Z28

Z48

� �!

Z20

Z44

� �!

Z30

Z36

� �!

Z15

Z42

� �! output:

To obtain the whole course from the appearance of tooth crack to tooth breakage, signalswere sampled constantly. An accelerometer mounted on the outer surface of the outputshaft bearing housing was used to acquire the vibration signals. The reason for selecting

4847

3630 22

17

28 3438 38

4443

36

30

15

42

45

Input

20

Output

Figure 2. The transmission diagram of an automobile transmission box.


this spot to collect vibration signals is that the gear pair Z15=42 is the slowest and thus,endures the largest torque. Fatigue crack on gear tooth would appear first on this pair.

The rotating speed of the input shaft was 1600 r/min, i.e. 26.67Hz. The rotatingfrequency of the output shaft was 2.1Hz. The faulty gear was discovered to be the 42-tooth gear on the output shaft. Cracks appeared in two symmetrical teeth. The signals inthis case should include the impulse components whose period equals 0.24 s. The signalswere sampled at 5KHz with a 2KHz low-pass filter in advance. A few minutes before thetooth breakage, fatigue cracks were discovered. The collected signals should includeperiodic impulses due to crack.

Figure 3 shows the waveform with broken teeth. The waveform has clear impulses.Engineers usually try to detect cracks to prevent gear tooth breakage. There is no muchuse to analyse signals with broken teeth. On the other hand, Fig. 4 shows the signalscollected before any tooth was broken. No periodic impulses appear on the waveform inFig. 4 even though cracks may have appeared in the gear already. The periodic impulses ofthe cracks were hidden in the signals. Two different types of orthogonal wavelets were usedto perform DWT for the signals shown in Fig. 4. Figure 5 shows the decimated wavelettransform result using DB4 wavelet. Figure 6 shows the undecimated wavelet transformresult using harmonic wavelet proposed by Newland [13]. No obvious periodic impulsesappear in Fig. 6. In Fig. 5, two sharp impulses appear in the subplot of scale 24 and in thesubplot of scale 22, while no impulses exist in other subplots. As a result, it is difficult todraw any conclusive result from either Fig. 5 or Fig. 6. As will be revealed later with theproposed adaptive wavelet filter, there are altogether three impulses in the signal.

4. GEAR TOOTH CRACK RECOGNITION USING ADAPTIVE WAVELET FILTER

As introduced in Section 2, Morlet wavelet is used to obtain the adaptive wavelet filter.Let parameter b vary from 0.1 to 4 with a step size of 0.1, the scale vary from 1 to 30 with a

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2000

-1500

-1000

-500

0

500

1000

1500

2000

Time: Second

Vol

tage

: m

V

Figure 3. The waveform of the gearbox with a broken tooth.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2000

-1500

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500

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Figure 4. The waveform of the gearbox with tooth crack.

Figure 5. The DWT of the signals in Fig. 4 with the DB4 wavelet.


step size of 1. The largest kurtosis value of 8.8 is obtained when b ¼ 0:3 and the scaleequals to 19, as shown in Fig. 7. To illustrate the sensitivity of the kurtosis as a function ofb, we fix the value of the scale to be 19 and plot the kurtosis–b relationship in Fig. 8, whichshows that the kurtosis is very sensitive to the value of b. The filtering result with the

Figure 6. The DWT of the signals in Fig. 4 with the harmonic wavelet.

Figure 7. The kurtosis dsitribution for different values of b and the scale.


optimised wavelet filter (b ¼ 0:3 and a ¼ 19) is shown in Fig. 9. Periodic impulses areobvious in Fig. 9. The period is just about 0.24 s.

To demonstrate the efficiency of the proposed adaptive wavelet filter, two other valuesof parameters a and b corresponding to relative large kurtosis are selected to produce twodifferent wavelet filters. One set of parameter values is a ¼ 12 and b ¼ 0:1 and thecorresponding kurtosis value is equal to 7.9. The results with this wavelet filter are shown

Figure 8. The kurtosis as a function of b when the scale a ¼ 19.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2000

-1500

-1000

-500

0

500

1000

1500

2000

Time: Second

Vol

tage

: mV

Figure 9. The adaptive filtering result for the vibration signals of the gearbox with tooth crack.


in Fig. 10. There is still heavy noise in the waveform shown in Fig. 10. Only two impulsesappear in Fig. 10 and the impulses look obscure. Another set of parameter values is a ¼ 27and b ¼ 4 and the corresponding kurtosis value is equal to 8.0. The results with thiswavelet filter are shown in Fig. 11. In Fig. 11, the three impulses are clearer because the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

500

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1500

2000

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-500

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-1500

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tage

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Figure 10. The filter result when b ¼ 0:1 and the scale a ¼ 12 for the wavelet filter.

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8

Time: Second

0

500

1000

1500

2000

-500

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tage

: mV

Figure 11. The filter result when b ¼ 4 and the scale a ¼ 27 for the wavelet filter.


kurtosis value is larger than the last one. Compared with the waveform shown in Fig. 9,the impulses in Fig. 11 have smaller time duration. This shows that the wavelet filterobtained with the largest kurtosis value provides the best filtering results.


5. CONCLUSIONS

Wavelets are essentially octave filters. Wavelet decompositions amount to performingfiltering for signals. Researchers and engineers are used to performing this operation usingMallat or a trous algorithm while seldom conducting optimisation on the filter. Theadaptive wavelet filter proposed in this paper represents an attempt in the direction ofoptimising the filter.

Morlet wavelet can be used for impulse detection due to its similarity to an impulse. Anydaughter wavelet can be viewed as a filter. To identify impulses hidden in noisy signals byfiltering, the daughter wavelet should match the time–frequency structure of the impulsewell. An adaptive Morlet wavelet filter based on kurtosis maximisation is proposed todetect periodic impulses automatically for recognition of gear tooth fatigue crack.Compared with other orthogonal wavelet decompositions, the result obtained by thismethod is more effective for capturing impulses.

ACKNOWLEDGEMENTS

The vibration data used in this study were provided by the Research Insitute ofDiagnostics & Cybernetics, Xi’an Jiaotong University, P.R. China. Support from NaturalSciences and Engineering Research Council of Canada is acknowledged.

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gearbox fault diagnosis using adaptive wavelet filter

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